As for a double pyramidal central configuration in 6-body problems, the case when its base is a concave polygon is studied. By advancing several assumptions according to the definition of double pyramidal central configuration and deducing two theorems and two corollaries on this subject, the essential and sufficient conditions to form a double pyramidal central configuration with a concave quadrilateral base are demonstrated.
Based on some necessary conditions for double pyramidal central configurations with a concave pentagonal base, for any given ratio of masses, the existence and uniqueness of a class of double pyramidal central configurations with a concave pentagonal base in 7-body problems are proved and the range of the ratio between radius and half-height is obtained, within which the 7 bodies involved form a central configuration or form uniquely a central configuration.
In this paper, we give a short proof for the existence of nontrivial choreography solution to the equal-mass three-body problem, which is discovered by Chenciner and Montgomery recently.
